Acta Biomaterialia 12 (2015) 93–101

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Acta Biomaterialia

journal homepage: www.elsevier .com/locate /actabiomat

Geometric characterization and simulation of planar layered elastomericfibrous biomaterials

http://dx.doi.org/10.1016/j.actbio.2014.09.0491742-7061/� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: W.A. ‘‘Tex’’ Moncrief, Jr. Simulation-Based Engineer-ing Science Chair I, Institute for Computational Engineering and Sciences (ICES), TheUniversity of Texas at Austin, 201 East 24th Street, ACES 5.438, 1 University Station,C0200, Austin, TX 78712-0027, USA. Tel.: +1 512 232 7773; fax: +1 512 471 8694.

E-mail address: msacks@ices.utexas.edu (M.S. Sacks).

James B. Carleton a, Antonio D’Amore b, Kristen R. Feaver a, Gregory J. Rodin a,c, Michael S. Sacks a,⇑a Center for Cardiovascular Simulation, Institute for Computational and Engineering Sciences, Department of Biomedical Engineering, University of Texas at Austin, Austin,TX 78712, USAb McGowan Institute for Regenerative Medicine, University of Pittsburgh, Pittsburgh, PA 15261, USAc Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, TX 78712, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 May 2014Received in revised form 28 August 2014Accepted 28 September 2014Available online 13 October 2014

Keywords:ScaffoldsElectrospunFiber geometrySimulationMicromechanics

Many important biomaterials are composed of multiple layers of networked fibers. While there is a grow-ing interest in modeling and simulation of the mechanical response of these biomaterials, a theoreticalfoundation for such simulations has yet to be firmly established. Moreover, correctly identifying andmatching key geometric features is a critically important first step for performing reliable mechanicalsimulations. The present work addresses these issues in two ways. First, using methods of geometricprobability, we develop theoretical estimates for the mean linear and areal fiber intersection densitiesfor 2-D fibrous networks. These densities are expressed in terms of the fiber density and the orientationdistribution function, both of which are relatively easy-to-measure properties. Secondly, we develop arandom walk algorithm for geometric simulation of 2-D fibrous networks which can accurately repro-duce the prescribed fiber density and orientation distribution function. Furthermore, the linear and arealfiber intersection densities obtained with the algorithm are in agreement with the theoretical estimates.Both theoretical and computational results are compared with those obtained by post-processing of scan-ning electron microscope images of actual scaffolds. These comparisons reveal difficulties inherent toresolving fine details of multilayered fibrous networks. The methods provided herein can provide arational means to define and generate key geometric features from experimentally measured or pre-scribed scaffold structural data.

� 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Many important biomaterials are composed of multiple layersof networked fibers. Elastomeric fibrous scaffolds used in engineer-ing soft tissues are a prime example [1]. Since soft tissues undergolarge deformations [2,3], the constituent fibers must have elasto-meric characteristics and undergo large macroscopic deformationsas a result of large rotations and strains. These characteristics allowthe scaffolds to duplicate many of the salient characteristics of thesoft tissues they are intended to replace [4]. The scaffolds are alsoexpected to have multiple functions, including facilitation of cellattachment and support of de novo tissue synthesis and stresstransfer [1]. To meet these multifaceted demands, one must

develop a fundamental understanding of the underlying physicalprocesses occurring within the scaffolds across multiple scales [1].

Currently, electrospinning is a common process for elastomericscaffold fabrication [1,5]. This process results in very long, oftenundulated ‘‘continuous’’ fibers that form dense networks with lay-ered structures. Other approaches focus on reconstituted collagenor fibrin gels [6]. Regardless of the methodology used to createthe scaffold structures, it is well established that the fiber geome-try significantly influences the macroscopic mechanical response[7,8]. Accordingly, there has been a considerable amount of effortdevoted to characterization of the fibrous geometry, with the focuson measuring an in-plane fiber orientation density function andestablishing its relations with mechanical anisotropy [5,9–13].

Generally, current approaches to simulating fibrous networkgeometries are based on: (i) generating Voronoi diagrams [14];(ii) generating assemblies of straight fibers with prescribed orien-tations [15]; (iii) using method (ii) to generate a network that isused as the initial state of a dynamics simulation in which the fiberpositions and orientations evolve, and crosslinks are formed [16];

94 J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101

and (iv) direct post-processing of scaffold images [10]. While allfour approaches have been successfully used for predicting themacroscopic mechanical response in terms of fiber stiffness andvolume fraction, the generated geometries simply do not consis-tently resemble the actual ones and involve somewhat subjective,empirical rules. Most importantly, the geometric parameters nec-essary for modeling and simulation of other aspects of mechanicalbehavior of fibrous networks are yet to be defined.

Herein, we rely on methods of geometric probability to developtheoretical estimates for the mean linear and areal fiber intersec-tion densities for 2-D fibrous networks. These densities areexpressed in terms of the fiber density and the orientation distri-bution function, both of which are relatively easy-to-measureproperties. Secondly, we develop a 2-D random walk algorithmcapable of generating realistic geometric models of fibrousnetwork layers. This algorithm can greatly benefit further studiesconcerned with micromechanical modeling by generating geome-tries that resemble actual ones. We further present simulationresults aimed at validating the assumptions underlying thetheoretical development, testing the algorithm performance andestablishing the minimum specimen size necessary for capturingmacroscopic properties. Finally, we show how the theory and sim-ulations can complement the use of experimental observations tomore accurately determine important features of the material’smicrostructural geometry.

2. Methods

2.1. Derivation of key geometric relationships for planar fibrousnetworks

The materials considered in the present work can be idealizedas a set of multiple, connected planar 2-D networks of long, curvedfibers. This has been shown to be a very good idealization for elec-trospun polyester urethane urea (ES PEUU) scaffolds (Fig. 1), aswell as many other important biomaterials. Our approach is thusrestricted to layered networks, which can be treated as an assem-bly of 2-D monolayers. We introduce basic quantities that charac-terize 2-D fibrous geometries, the linear and areal intersectiondensities for planar fibrous networks, and develop simple expres-sions in terms of the fiber density and orientation density function.These quantities allow estimation of a characteristic segmentlength between neighboring fiber intersections. The derivedexpressions significantly simplify the task of microstructuralcharacterization, since the intersection densities are difficult tomeasure directly.

We begin with a simplified model, in which a fibrous layer isrepresented as a 2-D network the building blocks of which areshort, straight line segments. First, we establish some results forsuch networks of disjoint segments, then we extend those results

Fig. 1. Scanning electron microscope image of the top layer of an ES PEUU scaffold[10]. The fiber diameter is approximately 1 micron.

to networks of long, curved fibers composed of connected seg-ments. In the process, we adopt minimal assumptions requiredto establish simple expressions for the linear and areal intersectiondensities in terms of the linear fiber density and the orientationdensity function (ODF). The mathematical problem considered inthis section can be classified as a generalization of the classical Buf-fon’s needle problem [17]. In this problem, a needle of knownlength is dropped onto a floor composed of many parallel floorboards of known width, and we seek the probability that the nee-dle lands on a boundary line between the adjacent floorboards.This problem represents the beginning of the field of geometricprobability [18].

Let us consider a domain in Euclidean space X 2 R2, the area ofwhich is equal to A. The domain contains m straight line segments,each of length s. We suppose that m� 1 and assign to the seg-ments an ODF f ðuÞ such that the probability that any given seg-ment has an orientation between u and uþ Du is Dp ¼ f ðuÞDu.The ODF is periodic, such that f ðuÞ ¼ f ðuþ pÞ, and therefore theangle u can be restricted to the interval ½0;pÞ. The segment centersare uniformly distributed in X following a Poisson’s process [19]. Itis expedient to assume that X is a unit cell of a periodic structure.Accordingly, the segments protruding outside of X can re-enter itfrom the opposite side.

Let us consider a randomly placed test straight line segment Tof length Lt cutting through some of the m segments in X and ori-ented at an angle u. Then a segment S, oriented at an anglebetween u0 and u0 þ Du0, intersects T if and only if the center ofS is located inside the parallelogram P with edges of length Lt

and s, and angle ju�u0j (Fig. 2).Since the area of P is equal to sLt sin ju�u0 j, the probability

that S is oriented at an angle between u0 and u0 þ Du0, and inter-sects T is equal to

Dp ¼ sLt sin ju�u0jA

f ðu0ÞDu0 ð1Þ

Accordingly, the probability that any segment, regardless of itsorientation, intersects T is equal to

pðuÞ ¼ sLt

A

Z p

0sin ju�u0jf ðu0Þdu0: ð2Þ

Let n denote the number of intersections along T ; the corre-sponding probability density function is denoted by gðnÞ. Sincethe position and orientation of each segment are independent ofthe positions and orientations of all the other segments, the inter-sections with T are independent events. This is a classical settingleading to the binomial distribution with the probability densityfunction

gðnÞ ¼ m!

n!ðm� nÞ! pnð1� pÞm�n ð3Þ

Fig. 2. An arbitrary segment S having length s and orientation angle u0 intersectsthe test segment T having length Lt and orientation angle u if its center lies withinthe parallelogram P designated by the dashed lines.

J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101 95

Using Eq. (2), we express the mean of n as

lnðuÞ ¼ mp ¼ msLt

A

Z p

0sin ju�u0jf ðu0Þdu0: ð4Þ

Define the linear intersection density along the direction u as

kðuÞ :¼ nðuÞLt

: ð5Þ

The mean for this random variable is determined from Eq. (4):

lkðuÞ ¼LA

Z p

0sin ju�u0jf ðu0Þdu0

¼ nZ p

0sin ju�u0jf ðu0Þdu0: ð6Þ

Here L ¼ ms is the total fiber length and

n :¼ LA

ð7Þ

is the linear fiber density. Thus Eq. (6) constitutes a simple expres-sion for the mean linear intersection density along the direction uin terms of the linear fiber density n and ODF f ðuÞ. Unfortunatelysuch an expression cannot be established for the standarddeviation and other higher-order moments, as their dependenceson m and s prohibit these two parameters from being combinedinto the single parameter L. Note that n is an easy-to-measuremacroscopic parameter and f ðuÞ is relatively straightforward toobtain from microscopic images, which makes Eq. (6) particularlyuseful.

Following Morton [20], one may regard the segments as thebuilding blocks of rectifiable curves forming the network, and thusextend Eq. (6) to the entire network. This passage is valid only whenthe process preserves statistical independence of the segments,which, for our purposes, is a prohibitively restrictive condition. Thisissue will be addressed in Section 3, where we use simulations toexamine implications of the statistical dependence. If the networkis isotropic, then f ðuÞ ¼ p�1 and Eq. (6) implies

lk ¼2np: ð8Þ

Very similar calculations can be carried out for the areal inter-section density q. Accordingly, we obtain

lq ¼12

n2Z p

0

Z p

0sin ju�u0jf ðu0Þf ðuÞdu0du: ð9Þ

By adopting n�1 as the characteristic length scale, we can nor-malize both lk and lq so that the quantities of interest becomedimensionless. Then Eqs. (6) and (9) imply, respectively,

lkðuÞ :¼ lkðuÞn�1 ¼

Z p

0sin ju�u0jf ðu0Þdu0 ð10Þ

and

lq :¼ lqn�2 ¼ 12

Z p

0

Z p

0sin ju�u0jf ðu0Þf ðuÞdu0du: ð11Þ

2.2. An algorithm for generating networks of fibers

In this section, we describe an algorithm for generating 2-D net-works of intersecting fibers possessing the desired geometric char-acteristics. In the proposed algorithm, the network is generated asa collection of random walks constructed so that the network pos-sesses the prescribed fiber density and an ODF. In the algorithm,each fiber is a piecewise linear curve generated as a random walkinvolving ns straight segments of the same length s, so that thelength of one fiber is Lf ¼ sns. The polar angles (orientations) of

the segments for a particular fiber are determined via the followingalgorithm.

1. Choose an origin O and a polar angle u0; as in the previous sec-tion u0 2 ½0;pÞ.

2. Construct the first segment such that its origin is at O, and itdeviates from the direction u0 by an angle h1, which is a reali-zation of a random variable uniformly distributed over theinterval �a � h1 � a with 0 < a � p; note that the polar angleof the first segment is u1 ¼ u0 þ h1.

3. Continue the process such that the polar angle of the kth seg-ment is uk ¼ uk�1 þ hk, where hk is a realization of a randomvariable, uniformly distributed over the interval ½�a;a�.

Most frequently, random walks are constructed using a ¼ p.This case, however, is of little interest to us, as we are interestedin constructing fibers that do not depart dramatically from straightlines. For this purpose, a should be small.

According to the algorithm, the end-to-end vector of a fiberformed by ns segments can be prescribed by the complex number

z ¼ sXns

k¼1

expðiukÞ: ð12Þ

The mean of z is

lz ¼1

ð2aÞns

Z a

�a

Z a

�a:::

Z a

�azdh1dh2:::dhns

¼ s expðiu0Þbð1� bns Þ

1� bwith b ¼ sin a

a: ð13Þ

This equation implies that u0 is the expected polar angle of theend-to-end vector. Therefore a desired ODF f ðuÞ for a network canbe approximated by generating the initial angles u0 for the indi-vidual fibers from the ODF f ðu0Þ. This approximation requires asufficiently large number of fibers.

The algorithm is formulated in terms of dimensionless parame-ters, using n�1 as the normalizing length scale, as in Eqs. (10) and(11); the normalized parameters are denoted by the hat symbol(^). Then for X chosen to be a square box of size LB, the normaliza-tion implies the following relationship among the dimensionlessparameters:

nf nss

L2B

¼ 1: ð14Þ

The algorithm can be summarized as follows:

1. Prescribe a square LB � LB box, total fiber length L and ODFf ðuÞ.

2. Compute n from Eq. (7) and normalize all length scales bymultiplying them by n.

3. Select a sufficiently large nf for an accurate piecewise-con-stant approximation of f ðuÞ and generate nf initial anglesu0 consistent with this approximation.

4. Select a sufficiently small s ðs� LBÞ and compute ns from Eq.(14).

5. Select the random walk bounding angle a.6. Generate the network as a collection of nf random walks

each comprising ns segments.7. Compute the intersection points.8. Compute the actual ODF for the generated network.9. If the actual and prescribed ODFs are insufficiently close,

increase nf and/or decrease s until the two ODFs are suffi-ciently close.

10. Compute the densities kðuÞ and q.

96 J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101

In summary, to generate a network, one prescribes LB, L, andf ðuÞ and uses the parameters nf and s to control the accuracy,while the angle a is used as an adjustable parameter. The algorithmgenerates a network whose properties coincide with the prescribedones. Such a network consists of long piecewise linear fibers thatappear curved on the box scale (Fig. 3), thanks to the conditionLB � s. Note that, if desired, one can introduce splines to smoothenpiecewise linear random walks; this option is not pursued in thispaper.

To understand the role of the angle a, let us consider two net-works generated using identical parameters except for a (Fig. 4).Upon comparison of the generated networks, it becomes evidentthat small as result in almost straight fibers (Fig. 4a) whereas largeas result in tortuous fibers (Fig. 4b). We believe that quantificationof this geometric feature is essential for mechanical modeling ofnetworks. While we do not pursue this issue in this paper, weobserve that in principle the proposed algorithm allows one to con-trol tortuosity.

2.3. Representative volume elements

The determination of material properties must involve suffi-ciently large specimens, which properly represent the overall(macroscopic) material response. In the pertinent solid mechanicsliterature, such specimens are referred to as the representative vol-ume elements (RVEs), and we retain this terminology here despitethe fact that we are concerned with 2-D networks. A simple way oftesting if specimens of a particular size can be treated as RVEs is byexamining the statistics for a property of interest. For example, ifthe property of interest is the areal density of intersections q, then

Fig. 3. A representative simulated network of fibers: (a) fibers inside a periodic box;(b) close-up view showing the individual segments and their intersections.

Fig. 4. The effect of the angle a on network tortuosity: (a) a ¼ p64. (b) a ¼ p

8. Thenetworks were generated using L ¼ 50;000, LB ¼ 1000ðLB ¼ 50Þ, n ¼ 0:05, s ¼ 0:15,nf ¼ 1, ns ¼ 16;667, f ðhÞ ¼ 1=p.

all RVEs must yield essentially the same value of q. The term essen-tially means that q can have slight variations from one specimen toanother, but those variations must be small. In particular, one candemand that the coefficient of variation rq=lq corresponding to aset of specimens is small.

The notion of large specimens is meaningless unless one speci-fies a reference microstructural length scale. For 2-D networks, thisscale can be related to the mean areal density and set equal tol�1=2

q . For the purposes of dimensional analysis, the double integralin Eq. (9) can be set equal to unity, so that we obtain

LB � n�1 ) LB �L2

B

L) L� LB: ð15Þ

This equation implies that one can think of RVEs as specimensthat are densely filled with fibers.

3. Results

3.1. Statistical dependence of segments and minimum box size

In this section, we compare theoretical and simulation resultsusing the mean areal density lq as the quantity of interest. Weare also concerned with identifying the minimum RVE size neces-sary for treating lq as a material rather than specimen property. Incomparing computed and theoretical values, it is important to keepin mind that the algorithm does not generate statistically indepen-dent segments because the polar angle of each segment within afiber depends on the polar angles of the preceding segments,whereas the theoretical estimates are based on the assumptionthat the segments are statistically independent. Nevertheless, itis reasonable to expect that large numbers of long fibers tend togenerate samples of segments for which the theoretical estimatesshould be accurate. We thus present simulation results to confirmthis supposition.

For the first set of simulations, we selected ns ¼ 1, s ¼ 0:3 andf ðuÞ ¼ p�1; for ns ¼ 1, the value of a becomes irrelevant. The boxsize LB was treated as an independent variable and the value ofnf was computed from Eq. (14). The corresponding theoretical esti-mate follows from Eq. (11): lq ¼ p�1. Although this case is of littleinterest to the modeling of scaffolds, it allows us to generate statis-tically independent segments and examine how the computationaland theoretical results converge to each other as the network sizeincreases.

Computational results for lq vs. LB (Fig. 5) were obtained using10 realizations per box size. The error bars represent one standarddeviation for those samples. This plot shows that for LB � 100 thestandard deviation becomes very small and lq p�1. This suggeststhat the box size for computing macroscopic properties should beat least 100n�1; a more precise definition of the minimum RVE sizemust involve a prescribed error tolerance.

In the next set of simulations, we considered networks formedby long fibers. We selected a ¼ p=64, s ¼ 0:3 and f ðuÞ ¼ p�1, andadopted an additional condition of nss ¼ LB. Then Eq. (14) impliesLB ¼ nf , so that the network size is proportional to both ns and nf .As before, we conducted 10 simulations per box size. Simulationresults are compared with two theoretical predictions (Fig. 6).The first one (dashed line) is obtained by assuming thatf ðuÞ ¼ p�1, i.e. assuming an isotropic orientation distribution,and therefore lq ¼ p�1. The second theoretical prediction is moreaccurate, as it is obtained from Eq. (11) by integrating the actual,rather than the desired, ODF. It is clear that the latter predictionis in good agreement with simulation results even for LB 25.For LB � 100, simulation and theoretical results are very close,and the standard deviation for the simulation results becomessmall.

Fig. 5. The normalized areal intersection density lq vs. the normalized box size LB

for short fibers: The dots represent the mean values, the error bars represent onestandard deviation and the solid line is the theoretical estimate for the mean.

Fig. 6. The normalized areal intersection density lq vs. the normalized box size LB

for long fibers: The circles and the error bars represent the mean values andstandard deviations of the simulation results, the crosses represent the mean valuesderived from Eq. (11) using the actual ODF, and the dashed line represents the meanvalues derived from Eq. (11) using the uniform ODF.

Fig. 7. Prescribed (white) vs. actual (black) ODFs.

Fig. 8. Linear intersection density lkðuÞ: simulation (white), theory based on theactual ODF (grey), and theory based on the prescribed ODF (black).

J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101 97

3.2. Linear intersection density equation and non-uniform ODFs

In this section, we focus on the mean linear intersection densityand a non-uniform ODF. We selected a ¼ p=60, s ¼ 0:3, ns ¼ 100and nf ¼ 12; 000, so that Eq. (14) implies LB ¼ 600. The ODF wasprescribed as

f ðuÞ ¼ 30p

up

� �2ð1�u

pÞ

2ð16Þ

This ODF was assigned to the initial u0 and the actual ODF wasobtained based on the fibers constructed via the random walkalgorithm (Fig. 7). Both ODFs are represented as piecewise constantfunctions associated with uniformly partitioning the interval ½0;pÞinto 20 subintervals. The plot represents just one realization, butthe actual and prescribed ODFs are in fairly close agreement.

The linear intersection density lkðuÞ predicted from Eq. (10)based on the prescribed and actual ODFs was compared with thatobtained from the simulation (Fig. 8). The plot reveals that Eq. (10)is an excellent predictor of the simulation results (grey vs. whitebars). The small difference between the simulation and theoreticalresults based on the desired ODF (black vs. white bars) is explainedby the fact that the algorithm gives rise to slightly differentprescribed and actual ODFs.

3.3. Material design

While the algorithm can generate networks with desired proper-ties, the level and type of experimental control over these propertiesare very much technology dependent. Current electrospun technol-ogies, for example, can be adjusted to produce scaffold structureswith a wide range of geometric features, albeit in a trial-and-errorapproach. One example in the case of electrospun scaffolds is vary-ing the angular velocity of the mandrel to control the degree ofanisotropy (Fig. 9).

One of the advantages of the current method is that it is fabri-cation technology independent, so that such geometric featuresrequired to obtain the desired macroscopic properties can bedetermined a priori and independent of the choice of a specific fab-rication method, whether currently available or developed in thefuture. For example, as shown in Fig. 10, one can use the algorithmto generate both isotropic (a) and anisotropic (b) networks.

3.4. Comparison with experimental data

In this section, we present simulation results for networks withgeometries that were generated so as to match the geometriesmeasured from scanning electron microscope images of actualelectrospun scaffold fiber networks. Three different images werechosen, each showing a fiber network having an ODF with a differ-ent degree of anisotropy: (i) nearly isotropic; (ii) moderately

Fig. 9. The degree of scaffold anisotropy can be controlled in the electrospinningprocess by varying the mandrel velocity. (a) Nearly isotropic – 250 rpm. (b) Highlyanisotropic – 2300 rpm [5].

Fig. 10. The ODF of the fibers may be controlled by prescribing f ðu0Þ. The networkin (a) is isotropic, whereas the network in (b) has a preferred direction, as shown bythe arrows. Also, as shown in Fig. 4 one can use the angle a to generate networkswith desired tortuosity.

Fig. 11. Some of the image traces of the top layer of ES PEUU scaffolds by one of theanalysts. Each image shown is 1/6 of the total image area traced for each scaffold.The degree of anisotropy of the scaffolds was controlled by the angular velocity ofthe mandrel. (a, b) Nearly isotropic – 250 rpm. (c, d) Moderately anisotropic –750 rpm. (e, f) Highly anisotropic – 2300 rpm.

98 J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101

aligned; and (iii) highly aligned. For each of the three images, fiveanalysts were given the task of tracing the fibers in the top layerand marking the associated fiber intersections (Fig. 11). We thusobtained 15 total observations.

The total length of fiber traced was extracted using a skeleton-ization plugin for ImageJ [21]. The image height and width and theODF were measured using a custom MATLAB program [5]. Fromthis raw data, the fiber densityn, the ODF f ðuÞ and the normalizedareal intersection density lq were derived for all 15 observations.The five experimentally derived normalized intersection densitieslq are plotted (filled circles) for each of the three images in Fig. 12.

Looking only at these experimentally derived results for anyone of the images (Fig. 12(a), (b) or (c)), we can see that there isa large degree of variation in the measurements of the analysts.Since all five analysts were given identical copies of the image,we can conclude that this variation is due to differences amongthe analysts’ interpretations of the images.

Next, for each of the three images, we used the measured ODFf ðuÞ in Eq. (11) to determine the normalized intersection densitylq predicted by the theory presented in Section 2. The lq predictedusing the measured ODF is plotted (solid line) for each image inFig. 12. The first conclusion that may be drawn is that the predictedlq decreases with increasing alignment, as can be seen by compar-ing Fig. 12(a), (b) and (c). Also, by comparing the predicted lq

(solid line) and measured values (filled circles), we notice that, inaddition to the large degree of random variation between themeasurements of the different analysts, there is also a systematicdeviation between the measured and predicted values.

In order to determine whether part of this discrepancy may beattributed to specimen size, we performed simulations using largeand small specimens. The measured values of n and f ðuÞwere usedas inputs to the algorithm developed in Section 2 to generate thesimulated geometry. We selected a ¼ p=60, s ¼ 0:3 and ns ¼ 100

so that Eq. (14) implies L2B ¼ 30nf . For each of the 15 architectures,

we generated two sets of 10 realizations. For the first set of realiza-tions, we used the very large specimen size LB ¼ 300. For thesecond set, we used much smaller sizes, which were chosen tomatch the specimen sizes of the experimentally measured images.These smaller specimen sizes fell in the range 16 � LB � 46. Themean and standard deviation of lq from the large-specimen(unfilled squares) and small-specimen (unfilled circles) sets of real-izations are shown in Fig. 12. The error bars represent one standarddeviation for those samples. For each microstructure, we also usedEq. (11) to calculate lq predicted by the theory using the f ðuÞ actu-ally generated by the algorithm, rather than the measured f ðuÞthat the algorithm was targeting. The means of these predicted val-ues are shown for the large-specimen (xs) and small-specimen(crosses) sets of realizations in Fig. 12.

For the large specimen simulations, the mean lq values (opensquares) coincide almost exactly with the theoretically predictedvalue (solid line), and the error bars are so small that they arebarely visible. We can thus assume that this size is sufficient fortreating the specimens as RVEs. For the small specimen simula-tions, there is a large degree of variation in the mean lq values(open circles), and the error bars are large. This variation can beunderstood by noticing that the mean values coincide almostexactly with theoretical predictions based on the ODF actuallyproduced by the algorithm (crosses). Therefore, the specimen-to-specimen variation in lq can be explained by the inability of thealgorithm to correctly reproduce the measured ODF when thespecimen size is small. However, the systematic deviation between

Fig. 12. lq for each analyst: the filled circles are the experimental observations.Unfilled circles and squares are the mean values of the simulations, and the errorbars represent one standard deviation. The crosses are values derived from Eq. (11)using the actual ODF, and the line represents the values derived using theexperimentally measured ODF.

J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101 99

the measurements (filled circles) and the theoretical prediction(solid line) cannot be attributed to the small specimen size causinga mismatched ODF, since the measured ODF was used in Eq. (11) tomake the prediction. This suggests that the systematic deviation islikely caused by other factors.

4. Discussion

4.1. Overview

The future state of the art in biomaterial fabrication could allowa degree of finely controllable microstructural design not possiblewith current non-woven elastomeric biomaterials. Thus, the devel-opment of computational frameworks that allow for the design ofmacroscopic properties based on microscopic features representsa positive departure from what has mostly been an empirically

driven field. Central to this objective is elucidating how the uniquestructural characteristics exhibited by these materials translate todistinct macroscopic mechanical properties, especially in relationto duplicating specific biomechanical tissue properties. It shouldbe noted that, while the elastomeric fiber networks of electrospunscaffold materials are of particular interest to the biomaterials com-munity, the applicability of the theory and algorithm presentedhere is not restricted to the elastomeric case. The elastomeric case,in which the material experiences large deformations at the macro-scale, is one of the more difficult to work with since the fibers willexperience large rotations at the microscale. A wide variety ofimportant biomaterials may be modeled, such as collagen fibernetworks in skin and connective tissue, cellular stress fibers andbasement membranes. The current approach is well suited to anylayered material composed of long intersecting fibers.

The current work presents approaches for characterizing andsimulating layered fibrous materials, based on theoretical estimatesfor the mean linear and areal intersection densities for 2-D fibrousnetworks. It builds on the experience with periodic elastic lattices[22,23], which share some geometric and mechanical features withfibrous networks. The macroscopic elastic properties of a periodicelastic lattice are derived from a unit cell analysis based on classicalEuler–Bernoulli beam theory. Networks generated using the algo-rithm described in Section 2 can be used as the mesh for finite ele-ment simulations of multiscale mechanical behavior of layeredfibrous materials. The current theoretical results are valuablebecause they allow one to estimate the difficult-to-measure inter-section densities, which have a significant effect on bulk mechani-cal behavior, in terms of the easy-to-measure linear fiber density nand the ODF. The algorithm is valuable because it can be adoptedfor micromechanical simulations of electrospun scaffolds and otherlayered elastomeric fibrous biomaterials. The theoretical andnumerical results compare favorably with each other for suffi-ciently large networks, which can be characterized as RVEs. Forthe mean areal intersection density it is sufficient to use specimenswith LB � 100n�1. The mean linear intersection density functionrequires higher values; LB � 600n�1 for the example presented inSection 3.

We note that the current theoretical development and algo-rithm are restricted to a single 2-D layer. Moreover, these theoret-ical and algorithmic developments are not easily extended to threedimensions. The key difficulty is that the intersection densities inthree dimensions depend not only on the fiber length but also fiberdiameter. However, the current algorithm may be easily extendedto generate 3-D geometries that consist of multiple connected lay-ers of 2-D networks. In this case, each sequential layer is generatedin a plane that is located one fiber diameter above the previouslayer. During the random walk process, intersections are generatednot only with segments within the same layer, but also with seg-ments in adjacent layers. This extended algorithm may be usedto generate realistic geometry for a broad range of layered fibrousnetwork materials, including the electrospun structures consid-ered in the present work. The algorithm may also be improved inorder to attain an even more accurate approximation of desiredODFs. The improvements could include adaptivity, so that the ini-tial angles u0 are not chosen a priori but rather using the currentODFs for the fibers that have been already generated.

The comparison between theoretical estimates and experimen-tal measurements shows that measuring network intersection den-sity lq unambiguously is very difficult (Fig. 12). The largesystematic and random deviations from the predicted values arenot all that surprising, even under ideal laboratory conditions. Inreal images of fiber networks, in which fibers have non-zero diam-eter and weave over and under each other, the determination of‘‘what counts as an intersection’’ and ‘‘which fibers are on the toplayer’’ becomes open to individual interpretation. In contrast, the

100 J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101

simulation results are very consistent because the geometry is ide-alized as a single layer of zero-thickness curves, so there is no ambi-guity related to fiber intersections or layers. While the linear andareal intersection densities of real networks are difficult to measuredirectly, these quantities may be determined with relative easeusing the relationships between microscopic and macroscopicquantities derived in this paper. In contrast with intersectiondensity lq, the fiber density n and ODF f ðuÞ of a network are mac-roscopic quantities and may be measured with high accuracy usingimages or approximated using knowledge of the manufacturingconditions. Subsequently, these quantities may be used in Eq. (6)and Eqs. (9–11) to determine the intersection densities of thesenetworks.

4.2. Limitations

The fibrous network geometry generated using the proposedalgorithm is a good idealization of many materials of interest tothe biomaterials community and represents a significant advancein the fidelity of geometric representations of such materials. How-ever, there are some limitations to the model. One limitation of ourapproach is that the fiber orientation in the initial configurationgenerated using the proposed algorithm is not permitted haveany component perpendicular to the plane. This is not a significantlimitation, however, since the proposed stacked layer approach is avery good approximation for electrospun scaffolds, as well as manytissues such as basement membranes. In such scaffolds, most fiberout-of-plane undulation amplitudes are not more than a few fiberdiameters. In our approach, fibers form intersections with fibers inadjacent layers, which are separated by a distance of one fiberdiameter, so this is a reasonable, albeit imperfect, representationof the real material. Additionally, within each layer, the fibers arerepresented as 1-D curves that reside strictly in the 2-D plane, soeach fiber intersection contains a single point that is shared bytwo curves. Of course, a real fiber has a non-zero diameter and can-not pass through other fibers in the plane, but must weave aboveand below them at a centerline-to-centerline distance of at leastone fiber diameter. Finally, we note that another characteristic ofthe proposed algorithm is that there are only a few input parame-ters to select, which makes the algorithm relatively easy to imple-ment and use. However, this could lead to a situation where onehas less control over the geometric characteristics of the resultingnetwork than could be gained by using a more complex algorithmwith more user inputs.

4.3. Relations to biomaterial scaffold micromechanics

The usefulness of the present findings can be best understood inthe context of an improved appreciation of how electrospun scaf-folds function across multiple length scales. Fiber tortuosity, ameasure of how much a fiber deviates from being straight in theunloaded scaffold, was observed to be dependent on mandrelvelocity during both production and orientation. As the scaffoldunderwent planar biaxial modes of deformation, fiber tortuositywas extinguished and substantial fiber rotational kinematics wasobserved to contribute to an intricate fiber recruitment process[24]. Electrospun constructs were observed to follow gross affinefiber transformations. Interestingly, some fibers were observed torotate or change their direction of orientation during deformationwhile, as a population, no net change was measured. Furthermore,it was observed that neighboring fibers were well attached wherethey overlapped or intersected. These attachments impede transla-tion of fibers with respect to one another but do not appear to inhi-bit rotational fiber kinematics about these points of intersection.The overall strain behavior tends to become increasingly morehomogeneous as the scale of interest approaches the tissue level.

In a recent study, however, elastomeric electrospun constructfabrication conditions were evaluated with regard to their effecton fiber geometry and biaxial mechanical behaviors [11]. It wasdetermined that fiber alignment alone was insufficient to predictmechanical response.

Simulation clearly plays an important role in understandingthese phenomena [25]. The underlying models must be based ona necessary subset of morphology descriptors, such as materialporosity, fiber density, fiber alignment distribution, fiber connec-tivity distribution and fiber diameter, which may be measuredthrough image analyses. The need for rigorously defined materialstructural descriptions has been extensively highlighted in recentliterature, where the model capability relies prevalently on theaccuracy of the network topology [26–30]. The present approachpresents a method to simplify such studies not only by defining alimited set of parameters, but also by defining such parametersthat are easily determined experimentally.

4.4. Conclusions and future work

The present algorithm can greatly inform the development oflayered fibrous network materials. While maturing, current effortsin the field of tissue engineering continue to be confounded by alack of understanding of the intricate multiscale hierarchicalarrangements and interactions typically found in native tissues.The current state of the art in biomaterial processing enables adegree of controllable microstructure that can be used for thedevelopment of model systems to deduce fundamental biologicalimplications of scaffold matrix morphologies on cell function. Fromour initial finite element simulations, one of the more interestingphenomena we have observed is the emergence of a long-rangefiber structure [25]. The precise relations between the initial net-work geometry and the resulting fiber kinematics under loadingremain to be elucidated. The work of understanding these relationsusing finite element simulations with meshes generated using thealgorithm presented here is the next phase in progressing towardthe ultimate goal of designing better materials. The first step willbe to validate the model by attempting to accurately reproducemacroscopic experimental stress–stretch curves of scaffolds underin-plane biaxial tension. Next, we will try to quantify therelationships between the geometric quantities proposed here tocharacterize the fiber network, and the shapes of the in-planestress–stretch curves. It may also be necessary to define additionalgeometric quantities to characterize the mesoscale structures thatemerge during loading. The 3-D nature of our simulated networkswill then allow us to investigate out-of-plane shear, bending andtwisting load cases. Other future work in the area of tissue engi-neering will include the addition of cellular and extracellularmatrix phases to the model, and using parallelization and multi-scale modeling approaches to model an entire heart valve leaflet.Ultimately, the development of computational frameworks thatenable the simulation of experimentally derived observationsrepresents a positive departure from what has mostly been anempirically driven field.

Acknowledgements

This work was supported by the National Institutes of Healthunder NIH R01 HL089750 and NIH R01 HL68816, and by theDepartment of Energy (National Nuclear Security Administration)under Award No. DE-FC52-08NA28615. The authors alsoacknowledge the Texas Advanced Computing Center (TACC) atthe University of Texas at Austin for providing HPC resources thathave contributed to the research results reported in this paper.URL: http://www.tacc.utexas.edu.

J.B. Carleton et al. / Acta Biomaterialia 12 (2015) 93–101 101

Appendix A. Figures with essential colour discrimination

Certain figures in this article, particularly Figs. 3, 4 and 10, aredifficult to interpret in black and white. The full colour imagescan be found in the on-line version, at http://dx.doi.org/10.1016/j.actbio.2014.09.049.

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